Statement: If f is differentiable on [a,b], then f' satisfies the intermediate value property: for any value k between f'(a) and f'(b), there exists c in (a,b) with f'(c) = k.

Implication: f' cannot have jump discontinuities. If f' is discontinuous, it must be an oscillatory (essential) discontinuity.

This is why: The derivative of $x^2$ sin$\frac{1}{x}$ at x = 0 exists (= 0), but f' oscillates near 0. The discontinuity is oscillatory, not a jump — consistent with Darboux's theorem.

JEE Application: If told f' has a jump discontinuity at some point, conclude that f is NOT differentiable there (since a derivative can't have jumps).